I ran across this video yesterday, by a math major, taking about other peoples reactions to learning that she is a math major.
I particularly like the analogy around 1:15 where she is talking about the jarring nature of switching from ordinary conversation to math. She likens it to being asked mid-conversation to compose a poem in Russian when you dont know Russian. I think that is a lovely analogy. She notes that math feels this way when you are not used to it, and sometimes even when you are used to it.
I think this is an apt analogy, because academic math comes out of left field for most people. In math class, it isnt that weird to have someone tell you:
The track at Made Up School is one mile long and features semi-circular ends connected by straight lines. Find the area enclosed by the track as a function of the radius of the semicircles. What dimensions allow the maximum area to be enclosed by such a track?
Say what? What does it look like? If the track is there already, how can we change it? Why are we doing this? And sometimes things get even worse:
If line segment BD is a perpendicular bisector of line segment AC, prove that triangle ABC is isosceles.
It just makes your brain hurt due to the sheer number of technical terms, and I have no sense whatsoever of this being a meaningful task that there would be a human reason for being able to do.
Notice that this is very different from other subjects that we study in school. In history, you might be asked:
In 1938, British Prime Minister Neville Chamberlain chose to adopt a policy of appeasement toward Hitlers aggression against Czechoslovakia. What did this mean? (from this list of sample questions)
There may be some confusing terms in here. Maybe Im not totally sure what a Prime Minister is, or who Chamberlain was, or where Czechoslovakia was in 1938, but I can get the sense of the question and have some idea of why I might want to be able to answer it, namely because I want to understand how the international world works.
Same with a thick subject like physics:
A battery is connected to a light bulb with copper wire to complete a circuit. The bulb immediately lights. Why?
Whoa, thats intense. You are asking how a battery works. It may seem like a hard thing to understand or explain, but I can see why I would want to try an answer it because I want to know how the world of electricity works.
I am not suggesting that there is no point to learning higher mathematics beyond arithmetic, but I am suggesting that those reasons can be obscure and subtle. We learn mathematics past basic computation because we want to understand the world, but it is an understanding of the world of thought, the world of algorithm, the world of logic, the world of abstraction. It is not the real world that we are seeking to understand, although higher mathematics often does have applications in the real world. Instead it is a fantasy world in which we ask what if and try to find a way to get consistent results. It is a world that is jarring precisely because it is so headily academic and is tethered to everyday concerns like a balloon that may slip away.
I think that if we all realized that we currently have enough math to understand our worlds, wed all be a lot happier. The math most of us use in life is more straight-forward than it is portrayed in school, and you may, right now, be as good at it as you need to be. Or you may find that you have some math-related problems in your real life that always frustrate you. That might be because they are really hard problems, and would be hard even for someone with advanced mathematical training. For instance, if you want to figure out a system of bonuses for your employees that reward certain types of job performance, then you probably will want to use some math, but the problem wont be simple, and math will only be one part of the solution.
I also love the end of the video above where Sarah emphasizes practice, and the fact that mathematical skills can be developed. Absolutely true. You probably already have most of the math skills that you need, and if you need more, practice is a good way to get more. Of course, one of the big troubles that I see is that K-16 math classes dont give people skills they will need after school, and it is actually quite hard to find needed and useful math skills if you arent in a STEM field (see, for instance, Audrey Watters on the difficulty of learning to code).